Animation flipped 180 degrees in Unity with .fbx export
Well it looks like it's caused by the existence of LocRotScale keyframes in Blender, i.e. keyframes related to the objects rather than the bones. It can be cleaned up either in Blender if possible, or those keyframes can be removed in Unity from the animation window after import.

Also, exporting with FBX version 6.1 instead of 7.4 from Blender seems to eliminate the flipping problem regardless, but not the origin problem.